Periodic Orbits and Homoclinic Orbits of Second-order Hamiltonian Systems with Mild Superquadratic Growth

Abstract

Using the saddle point theorem and the mountain pass theorem with Morse index estimate, the existence of periodic solutions for second-order Hamiltonian systems with mild superquadratic growth is proved in this paper. Meanwhile the existence result of homoclinic orbits for this kind of Hamiltonian systems is also obtained by the local convergence of a sequence of subharmonic solutions.

References

[1.] Abbondandolo A Morse Theory for Hamiltonian Systems, 2001 Boca Raton, FL Chapman & Hall/CRC.

[2.] Abbondandolo A, Molina J. Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems Calc. Var. Partial Differential Equations, 2000, 11(4): 395-430.

[3.] Ambrossiti A, Rabinowitz P. Dual variational methods in critical point theory and applications J. Functional Analysis, 1973, 14: 349-381.

[4.] Bartsch T, Szulkin A. Hamiltonian systems: periodic and homoclinic solutions by variational methods Handbook of Differential Equations—Ordinary Differential Equations, Vol. II, 2005 Amsterdam Elsevier B. V. 77-146

[5.] Chen G, Ma S. Periodic solutions for Hamiltonian systems without Ambrosetti–Rabinowitz and spectrum 0 J. Math. Anal. Appl., 2011, 379(2): 842-851.

[6.] Chen G, Ma S. Homoclinic orbits of superlinear Hamiltonian systems Proc. Amer. Math. Soc., 2011, 139(11): 3973-3983.

[7.] Coti Zelati V, Ekeland I, Séré E. A variational approach to homoclinic orbits in Hamiltonian systems Math. Ann., 1990, 228(1): 133-160.

[8.] Coti Zelati V, Rabinowitz PH. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials J. Amer. Math. Soc., 1991, 4(4): 693-727.

[9.] Ding Y, Willem M. Homoclinic orbits of a Hamiltonian system Z. Angew. Math. Phys., 1999, 50(5): 759-778.

[10.] Fei G., On periodic solutions of superquadratic Hamiltonian systems. Electron. J. Differential Equations, 2002, 2002: Paper No. 8, 12 pp.

[11.] Felmer PL, Silva E. Homoclinic and periodic orbits for Hamiltonian systems Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1998, 26(2): 285-301

[12.] Ghoussoub N. Location, multiplicity and Morse indices of min-max critical points J. Reine Angew. Math., 1991, 417: 27-76

[13.] Long Y. The minimal period problem of classical Hamiltonian systems with even potentials Ann. Inst. H. Poincaré C Anal. Non Linéaire, 1993, 10(6): 605-626.

[14.] Long Y. The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems J. Differential Equations, 1994, 111(1): 147-174.

[15.] Long Y. Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials Nonlinear Anal., 1995, 24(12): 1665-1671.

[16.] Long Y. On the minimal period for periodic solutions of nonlinear Hamiltonian systems Chinese Ann. Math. Ser. B, 1997, 18(4): 481-484

[17.] Mawhin J, Willem M Critical Point Theory and Hamiltonian Systems, 1989 New York Springer-Verlag.

[18.] Pipan J, Schechter M. Non-autonomous second order Hamiltonian systems J. Differential Equations, 2014, 257(2): 351-373.

[19.] Rabinowitz PH. Some critical point theorems and applications to semilinear elliptic partial differential equations Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1978, 5(1): 215-223

[20.] Rabinowitz PH Minimax methods in critical point theory with applications to differential equations, 1986 Providence, RI American Mathematical Society 65

[21.] Rabinowitz PH. Homoclinic orbits for a class of Hamiltonian systems Proc. Roy. Soc. Edinburgh Sect. A, 1990, 114(1–2): 33-38.

[22.] Séré E. Existence of infinitely many homoclinic orbits in Hamiltonian systems Math. Z., 1992, 209(1): 27-42.

[23.] Szulkin A, Zou W. Homoclinic orbits for asymptotically linear Hamiltonian systems J. Funct. Anal., 2001, 187(1): 25-41.

[24.] Tanaka K. Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits J. Differential Equations, 1991, 94(2): 315-339.

[25.] Tang C. Periodic solutions for nonautonomous second order systems with sublinear non-linearity Proc. Amer. Math. Soc., 1998, 126(11): 3263-3270.

[26.] Tang C, Wu X. Periodic solutions for a class of new superquadratic second order Hamiltonian systems Appl. Math. Lett., 2014, 34: 65-71.

[27.] Wang Z, Zhang J. Periodic solutions of a class of second order non-autonomous Hamiltonian systems Nonlinear Anal., 2010, 72(12): 4480-4487.

[28.] Ye Y, Tang C. Infinitely many periodic solutions of non-autonomous second-order Hamiltonian systems Proc. Roy. Soc. Edinburgh Sect. A, 2014, 144(1): 205-223.

[29.] Zhao F, Wu X. Saddle point reduction method for some non-autonomous second order systems J. Math. Anal. Appl., 2004, 291(2): 653-665.

[30.] Zhang Q, Liu C. Homoclinic orbits for a class of first order nonperiodic Hamiltonian systems Nonlinear Anal. Real World Appl., 2018, 41: 34-52.